{"id":"https://openalex.org/W7214722935","doi":"https://doi.org/10.48550/arxiv.2609.32970","title":"Obtaining Game-Stationary Points for Smooth Nonconvex-Nonconcave Minimax Problems via First-Order Methods","display_name":"Obtaining Game-Stationary Points for Smooth Nonconvex-Nonconcave Minimax Problems via First-Order Methods","publication_year":2026,"publication_date":"2026-09-26","ids":{"openalex":"https://openalex.org/W7214722935","doi":"https://doi.org/10.48550/arxiv.2609.32970"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.32970","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.32970","pdf_url":"https://arxiv.org/pdf/2609.32970","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.32970","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5111091300","display_name":"Mengxuan Dong","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I135310074","display_name":"University of Wisconsin\u2013Madison","ror":"https://ror.org/01y2jtd41","country_code":"US","type":"education","lineage":["https://openalex.org/I135310074"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Dong, Mengxuan","raw_affiliation_strings":["Department of Computer Sciences , University of Wisconsin-Madison"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Computer Sciences , University of Wisconsin-Madison","institution_ids":["https://openalex.org/I135310074"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5140268743","display_name":"Yao Yao","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I130238516","display_name":"University of Minnesota","ror":"https://ror.org/017zqws13","country_code":"US","type":"education","lineage":["https://openalex.org/I130238516"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Yao, Yao","raw_affiliation_strings":["School of Mathematics , University of Minnesota-Twin Cities"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics , University of Minnesota-Twin Cities","institution_ids":["https://openalex.org/I130238516"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5153458373","display_name":"Jiawei Zhang","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I135310074","display_name":"University of Wisconsin\u2013Madison","ror":"https://ror.org/01y2jtd41","country_code":"US","type":"education","lineage":["https://openalex.org/I135310074"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Zhang, Jiawei","raw_affiliation_strings":["Department of Computer Sciences , University of Wisconsin-Madison"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Computer Sciences , University of Wisconsin-Madison","institution_ids":["https://openalex.org/I135310074"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.5625,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.5625,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10545","display_name":"Optimization and Variational Analysis","score":0.38530001044273376,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.023800000548362732,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/first-order-stationary-points","display_name":"first-order stationary points","score":0.8640000224113464},{"id":"https://openalex.org/keywords/first-order-methods","display_name":"first-order methods","score":0.6230000257492065},{"id":"https://openalex.org/keywords/saddle-point-optimization","display_name":"saddle point optimization","score":0.42800000309944153}],"concepts":[{"id":"https://openalex.org/C149728462","wikidata":"https://www.wikidata.org/wiki/Q751319","display_name":"Minimax","level":2,"score":0.7688000202178955},{"id":"https://openalex.org/C189237950","wikidata":"https://www.wikidata.org/wiki/Q2500758","display_name":"Stationary point","level":2,"score":0.704800009727478},{"id":"https://openalex.org/C114466953","wikidata":"https://www.wikidata.org/wiki/Q6034165","display_name":"Initialization","level":2,"score":0.6456000208854675},{"id":"https://openalex.org/C203616005","wikidata":"https://www.wikidata.org/wiki/Q620495","display_name":"Hessian matrix","level":2,"score":0.6439999938011169},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6391000151634216},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.6312999725341797},{"id":"https://openalex.org/C2776330181","wikidata":"https://www.wikidata.org/wiki/Q18358244","display_name":"Maximization","level":2,"score":0.5370000004768372},{"id":"https://openalex.org/C50454189","wikidata":"https://www.wikidata.org/wiki/Q2075125","display_name":"Karush\u2013Kuhn\u2013Tucker conditions","level":2,"score":0.4699000120162964},{"id":"https://openalex.org/C18648836","wikidata":"https://www.wikidata.org/wiki/Q381892","display_name":"Compact space","level":2,"score":0.38830000162124634},{"id":"https://openalex.org/C28719098","wikidata":"https://www.wikidata.org/wiki/Q44946","display_name":"Point (geometry)","level":2,"score":0.37470000982284546},{"id":"https://openalex.org/C2776036281","wikidata":"https://www.wikidata.org/wiki/Q48769818","display_name":"Constraint (computer-aided design)","level":2,"score":0.35089999437332153},{"id":"https://openalex.org/C25343380","wikidata":"https://www.wikidata.org/wiki/Q277521","display_name":"Relation (database)","level":2,"score":0.3431999981403351},{"id":"https://openalex.org/C137836250","wikidata":"https://www.wikidata.org/wiki/Q984063","display_name":"Optimization problem","level":2,"score":0.32249999046325684},{"id":"https://openalex.org/C9652623","wikidata":"https://www.wikidata.org/wiki/Q190109","display_name":"Field (mathematics)","level":2,"score":0.296999990940094},{"id":"https://openalex.org/C22171661","wikidata":"https://www.wikidata.org/wiki/Q1074380","display_name":"Stochastic game","level":2,"score":0.29440000653266907},{"id":"https://openalex.org/C177067428","wikidata":"https://www.wikidata.org/wiki/Q17013331","display_name":"Feasible region","level":2,"score":0.28369998931884766},{"id":"https://openalex.org/C147764199","wikidata":"https://www.wikidata.org/wiki/Q6865248","display_name":"Minification","level":2,"score":0.2824000120162964},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.28110000491142273},{"id":"https://openalex.org/C63479239","wikidata":"https://www.wikidata.org/wiki/Q7353546","display_name":"Robustness (evolution)","level":3,"score":0.28060001134872437},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.2799000144004822},{"id":"https://openalex.org/C13662910","wikidata":"https://www.wikidata.org/wiki/Q193139","display_name":"Trajectory","level":2,"score":0.2766000032424927},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.27549999952316284},{"id":"https://openalex.org/C153258448","wikidata":"https://www.wikidata.org/wiki/Q1199743","display_name":"Gradient descent","level":3,"score":0.2671999931335449},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.26649999618530273},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.2581000030040741},{"id":"https://openalex.org/C63548660","wikidata":"https://www.wikidata.org/wiki/Q858223","display_name":"Limit point","level":2,"score":0.25679999589920044},{"id":"https://openalex.org/C132721684","wikidata":"https://www.wikidata.org/wiki/Q2877013","display_name":"Broyden\u2013Fletcher\u2013Goldfarb\u2013Shanno algorithm","level":3,"score":0.2533999979496002},{"id":"https://openalex.org/C55660270","wikidata":"https://www.wikidata.org/wiki/Q5164377","display_name":"Constrained optimization","level":2,"score":0.25290000438690186}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.32970","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.32970","pdf_url":"https://arxiv.org/pdf/2609.32970","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2609.32970","doi":"https://doi.org/10.48550/arxiv.2609.32970","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.32970","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.32970","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.32970","pdf_url":"https://arxiv.org/pdf/2609.32970","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7214722935.pdf","grobid_xml":"https://content.openalex.org/works/W7214722935.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W2970416783","https://openalex.org/W7027361433","https://openalex.org/W2954249541","https://openalex.org/W2905418717","https://openalex.org/W7133226754","https://openalex.org/W3207297452","https://openalex.org/W7213530129","https://openalex.org/W3006920537","https://openalex.org/W4414811188","https://openalex.org/W4320458645"],"abstract_inverted_index":{"Minimax":[0],"optimization":[1],"is":[2],"a":[3,54,64,81,98,111,123,127],"fundamental":[4],"framework":[5],"in":[6,91],"machine":[7],"learning,":[8],"robust":[9],"optimization,":[10],"and":[11,78],"game":[12,59],"theory,":[13],"yet":[14],"finding":[15],"first-order":[16,99,108,194],"stationary":[17,119,138,177],"points":[18,139],"of":[19,140,172,199],"general":[20],"nonconvex-nonconcave":[21,200],"minimax":[22,201],"problems":[23,202],"remains":[24],"challenging":[25],"without":[26],"additional":[27],"structural":[28],"assumptions.":[29],"Existing":[30],"guarantees":[31,196],"often":[32],"rely":[33],"on":[34,49,57,126,137],"global":[35,45,55],"PL-":[36],"or":[37,48,150,210],"KL-type":[38],"conditions":[39,51,85,158],"that":[40,52,101],"connect":[41],"max-player":[42,118],"stationarity":[43],"to":[44,63,89],"inner":[46,142],"optimality,":[47],"Minty-type":[50,211],"impose":[53],"relation":[56],"the":[58,71,141,160,164,173,206],"gradient":[60],"field":[61],"relative":[62],"reference":[65],"solution;":[66],"local":[67,112],"KL":[68],"variants":[69],"relax":[70],"former":[72],"requirement":[73,136],"but":[74],"typically":[75],"require":[76],"initialization":[77],"tracking":[79],"within":[80,106],"near-optimal":[82],"region.":[83],"Such":[84],"may":[86],"be":[87,148],"difficult":[88],"satisfy":[90],"many":[92],"applications.":[93],"In":[94,163],"contrast,":[95],"we":[96],"develop":[97],"method":[100],"finds":[102],"an":[103],"$\u03b5$-stationary":[104],"point":[105],"$\\widetilde{O}(\u03b5^{-2})$":[107],"iterations":[109],"under":[110],"inverse-Lipschitz":[113],"regularity":[114,189],"condition":[115,125,132],"around":[116],"approximate":[117],"points,":[120],"together":[121],"with":[122],"compactness":[124],"penalty":[128],"sublevel":[129],"set.":[130],"Our":[131],"places":[133],"no":[134],"optimality":[135],"maximization":[143],"problem:":[144],"they":[145],"need":[146],"not":[147,203],"globally,":[149],"even":[151],"locally,":[152],"maximizing.":[153],"We":[154],"further":[155],"provide":[156],"sufficient":[157],"for":[159,179,197],"required":[161],"regularity.":[162],"unconstrained":[165],"setting,":[166],"it":[167,184],"follows":[168,185],"from":[169,186],"uniform":[170],"nonsingularity":[171],"maximization-variable":[174],"Hessian":[175],"near":[176],"points;":[178],"constrained":[180],"upper":[181],"Moreau":[182],"envelopes,":[183],"standard":[187],"KKT":[188],"conditions.":[190],"These":[191],"results":[192],"establish":[193],"complexity":[195],"classes":[198],"covered":[204],"by":[205],"above":[207],"PL-,":[208],"KL-,":[209],"frameworks.":[212]},"counts_by_year":[],"updated_date":"2026-10-28T10:33:24.094632","created_date":"2026-09-30T00:00:00"}